De Morgan's Laws
~A⋂~B
~A⋃~B
Those should be lines over individual letters

Set Identities
~A⋂~B
~A⋃~B
Those should be lines over individual letters

Reverse De Morgan's Laws
~(A⋃B)
~(A⋂B)
Those should be lines over the whole top

Set Identities
A⋃(A⋂B)
A⋂(A⋃B)

Absorption Laws
A
A

Set Identities
A⋃~A
A⋂~A
Should be lines on top

Complement Laws
U
∅

Truth sets
{x|P(x)}

explain the parts

The elements of the set for which the values of x make the second part true - the elementhood test

y∈{x|P(x)}

P(y)

y∉{x|P(x)}

~P(y)

A is a truthset for the proposition P(x)
What is A in set form
What is meant by
y∈A
y∉A

P(y)
~P(y)

Redefine
{x∈U|P(x)}

{x|x∈U^P(x)}

Redefine
{x|x∈U^P(x)}

{x∈U|P(x)}

Redefine
y∈{x∈U|P(x)}

y∈U^P(y)

Redefine: What is it?
A⋂B

It is a set
{x|x∈A^x∈B}

If A and B have truth tests, then it could mean P(x)^Q(x)

What is it?
x ∈ A⋂B

It is a statement
x∈A^x∈B

and if they had propositions
P(x)^Q(x)

Redefine: What is it?
A⋃B

It is a set
{x|x∈A v x∈B}

If A and B have truth tests, then it could mean P(x) v Q(x)

What is it?
x ∈ A⋃B

It is a statement
x∈A v x∈B

and if they had propositions
P(x) v Q(x)

Redefine: What is it?
A/B

It is a set
{x|x∈A ^ x∉B}

If A and B have truth tests, then it could mean P(x) ^ ~Q(x)

What is it?
x ∈ A/B

It is a statement
x∈A ^ x∉B

and if they had propositions
P(x) ^ ~Q(x)

Redefine:
{x|x∈A^x∈B}

A⋂B

Redefine:
x∈A^x∈B

x ∈ A⋂B

Redefine:
{x|x∈A v x∈B}

A⋃B

x∈A v x∈B

x ∈ A⋃B

Redefine:
{x|x∈A ^ x∉B}

A/B

Redefine:
x∈A ^ x∉B

x ∈ A/B

What is it? Name it
Redefine:
A∆B

(2 definitions)

It's a set
Symmetric Difference
(A\B)⋃(B\A)
(A⋃B)\(A⋂B)

Redefine: what is it?

A⊆B
(2 definitions)

It is a statement:

∀x(x∈A → x∈B)
∀x∈A(x∈B)

Redefine: What is it?

A=B
(2 definitions)

It is a Statement

∀x(x∈A ↔ x∈B)
A⊆B ^ B⊆A

Redefine:
(A\B)⋃(B\A)
(A⋃B)\(A⋂B)

A∆B

Redefine:
∀x(x∈A → x∈B)
∀x∈A(x∈B)

A⊆B

Redefine:
∀x(x∈A ↔ x∈B)
A⊆B ^ B⊆A

A=B

What is it? What is it called?
Redefine
A⋂B=∅

A statement
Called disjoint
Allx, x is part of the first part iff x is part of the second part?
or This first part is a subset of the second, and the second is a subset of the first?

What is it?
Re-express:

{pi|i∈I}

An indexed set, the set contains all numbers pi with i being the element of some set.
{x|∃i∈I(x=pi)}

What is it?
Re-express:

{x|∃i∈I(x=pi)}

An indexed set, the set contains all numbers pi with i being the element of some set.

{pi|i∈I}

An indexed set, the set contains all numbers pi with i being the element of some set.

Give two ways

{x|∃i∈I(x=pi)}

{pi|i∈I}

Re-express the family of sets F in bracket notation:
using the classes of students Cs

give two ways

Each Cs is a set itself
{{1 2},{3,4},{5,6}} C1 = {1,2}

F = {Cs|s∈S}
{X|∃s∈S(X=Cs)}

What is it?
Re-express:

F = {Cs|s∈S}

Indexed Family
Each Cs is a set itself
{{1 2},{3,4},{5,6}} C1 = {1,2}

{X|∃s∈S(X=Cs)}

What is it?
Re-express:

F = {X|∃s∈S(X=Cs)}

Indexed Family
Each Cs is a set itself
{{1 2},{3,4},{5,6}} C1 = {1,2}

F = {Cs|s∈S}

What is it?
Re-express

P(A)

The power set:

{x|x⊆A}

x is a set

What is it?
Re-express

{x|x⊆A}

x is a set

The power set:
P(A)

The family set is a subset of what?
Give reasoning

If all courses were set C then each Cs is a subset of C

For each student Cs∈P(C)
Every element then of the family is an element of P(c) so F ⊆ P(C)

Equivalence

P(A⋂B)

P(A)⋂P(B)

Equivalence

P(A)⋂P(B)

P(A⋂B)

Explain and define:

⋂F

F = {{1,2},{2,3},{3,4}}
this is the intersection of the family
{1,2}⋂{2,3}⋂{3,4}
{x|∀A∈F(x∈A)}
{x|∀A(A∈F → x∈A)}

What is this?/

{x|∀A∈F(x∈A)}

F = {{1,2},{2,3},{3,4}}
this is the intersection of the family
{1,2}⋂{2,3}⋂{3,4}
{x|∀A∈F(x∈A)}
{x|∀A(A∈F → x∈A)}

What is this?

{x|∀A(A∈F → x∈A)}

F = {{1,2},{2,3},{3,4}}
this is the intersection of the family
{1,2}⋂{2,3}⋂{3,4}
{x|∀A∈F(x∈A)}
{x|∀A(A∈F → x∈A)}

What is this?

{1,2}⋂{2,3}⋂{3,4}

F = {{1,2},{2,3},{3,4}}
this is the intersection of the family
{1,2}⋂{2,3}⋂{3,4}
{x|∀A∈F(x∈A)}
{x|∀A(A∈F → x∈A)}

Explain and define:

⋃F

F = {{1,2},{2,3},{3,4}}
this is the union of the family
{1,2}⋃{2,3}⋃{3,4}
{x|∃A∈F(x∈A)}
{x|∃A(A∈F ^ x∈A)}

What is this?/

{x|∃A∈F(x∈A)}

F = {{1,2},{2,3},{3,4}}
this is the union of the family
{1,2}⋃{2,3}⋃{3,4}
{x|∃A∈F(x∈A)}
{x|∃A(A∈F ^ x∈A)}

What is this?

{x|∃A(A∈F ^ x∈A)}

F = {{1,2},{2,3},{3,4}}
this is the union of the family
{1,2}⋃{2,3}⋃{3,4}
{x|∃A∈F(x∈A)}
{x|∃A(A∈F ^ x∈A)}

What is this?

{1,2}⋃{2,3}⋃{3,4}

F = {{1,2},{2,3},{3,4}}
this is the union of the family
{1,2}⋃{2,3}⋃{3,4}
{x|∃A∈F(x∈A)}
{x|∃A(A∈F ^ x∈A)}

Define

⋂F=∅

undefined

What is it?
Redefine
⋂(i∈I) Ai

Where F puts Ai as sets in a set, this is more like the union of all the sets, it is the same thing as ⋂F
{x|∀i∈I(x∈Ai)}
{x|∀i(i∈I → x∈Ai)}
It asks that all the sets have the same element

What is it?

{x|∀i∈I(x∈Ai)}

⋂(i∈I) Ai
Where F puts Ai as sets in a set, this is more like the union of all the
sets, it is the same thing as ⋂F
{x|∀i∈I(x∈Ai)}
{x|∀i(i∈I → x∈Ai)}
It asks that all the sets have the same element

What is it?
Redefine
⋃(i∈I) Ai

Where F puts Ai as sets in a set, this is more like the union of all the sets, it is the same thing as ⋃F
{x|∃i∈I(x∈Ai)}
{x|∃i(i∈I ^ x∈Ai)}
That at least one of the sets of Ai have the element

What is it?
Redefine

{x|∃i∈I(x∈Ai)}

Where F puts Ai as sets in a set, this is more like the union of all the
⋃(i∈I) Ai
How is this different from a indexed set or indexed family? Because it's an element, not equal
sets, it is the same thing as ⋃F
{x|∃i∈I(x∈Ai)}
{x|∃i(i∈I ^ x∈Ai)}
That at least one of the sets of Ai have the element

What is it?
Define

AxB

Cartesian Product

{(a,b)| a∈A ^ b∈B}

results in all ordered pairs

What is it?
Define

{(a,b)| a∈A ^ b∈B}

Cartesian Product
AxB

results in all ordered pairs

Equivalence:
Ax(B⋂C)

(AxB)⋂(AxC)

Equivalence:
(AxB)⋂(AxC)

Ax(B⋂C)

Equivalence:
Ax(B⋃C)

(AxB)⋃(AxC)

Equivalence:
(AxB)⋃(AxC)

Ax(B⋃C)

Equivalence:
(AxB)⋂(CxD)

(A⋂C)x(B⋂D)

Equivalence:
(A⋂C)x(B⋂D)

(AxB)⋂(CxD)

Subset:

(AxB)⋃(CxD) ⊆

(A⋃C)x(B⋃D)

Equivalence:

Ax∅

∅xA
∅

AxB = BxA if and only if

A =∅ or
B = ∅ or
A=B

Truth set of P(x,y)

two versions

{(a,b)∈AxB | P(x,y)}
{(a,b) | (a,b)∈AxB ^ P(x,y)}

{(a,b)∈AxB | P(x,y)}

Truth set of P(x,y)

{(a,b) | (a,b)∈AxB ^ P(x,y)}

Truth set of P(x,y)

Define a relation from A to B

R ⊆ AxB
does not need to imply a truth set for R, it can be any subset

What is this?
R ⊆ AxB

A relation

Domain of R for AxB

{a∈A | ∃b∈B((a,b)∈R}

{a∈A | ∃b∈B((a,b)∈R}

Domain of R

Range of R

{b∈B | ∃a∈A((a,b)∈R}

{b∈B | ∃a∈A((a,b)∈R}

Range of R

Inverse of R R-1

{(b,a)∈BxA | (a,b)∈R}

{(b,a)∈BxA | (a,b)∈R}

Inverse of R R-1

Composition of two relations
R⊆AxB
S⊆BxC

SoR
{(a,c)∈AxC |
∃b∈B((a,b)∈R ^ (b,c)∈S}

{(a,c)∈AxC |
∃b∈B((a,b)∈R ^ (b,c)∈S}

Composition of two relations
R⊆AxB
S⊆BxC
SoR

(r,s)∈L-1 if and only if

(s,r)∈L

(s,p)∈ToE if and only if

∃c( (s,c)∈E ^ (c,p) ∈ T

(x,y) ∈ R

xRy

xRy

(x,y) ∈ R

Relation equivalence:
(R-1)-1

R

Relation equivalence:
Relation equivalence:Dom(R-1)

Ran(R)

Relation equivalence:
Ran(R)

Dom(R-1)

Relation equivalence:
Ran(R-1)

Dom(R)

Relation equivalence:
Dom(R)

Ran(R-1)

Relation equivalence:
To(SoR)

(ToS)oR

Relation equivalence:
(ToS)oR

To(SoR)

(SoR)-1

R-1 o S-1

Equivalence Relation:
R-1 o S-1

(SoR)-1

the relation R is reflexive on A (or just reflexive if A is clear from context) if